Try a finite example, and try to answer, "What is the sum of its elements cardinality?"
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Jonathan FischoffJul 20 '10 at 23:17

3

Intuitively, you can put many things in a basket -- but one thing you can never put in that basket is the basket itself.
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littleOAug 26 '14 at 14:55

1

The more one thinks about it, the more there is in it. The question should give a definition of set, and the answer should be based on the def itself, not on some other tall constructs.
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George ChenAug 28 '14 at 1:52

As others have explained, the existence of universal set is incompatible with the Zermelo–Fraenkel axioms of set theory. However, there are alternative set theories which admit a universal set. One such theory is Quine's New Foundations.

The "set of all sets" is not so much a paradox in itself as something that inevitably leads to a contradiction, namely the well-known (and referenced in the question) Russell's paradox.

Given any set and a predicate applying to sets, the set of all things satisfying the predicate should be a subset of the original set. If the "set of all sets" were to exist, because self-containment and non-self-containment are valid predicates, the set of all sets not containing themselves would have to exist as a set in order for our set theory to be consistent. But this "set of all sets" cannot exist in a consistent set theory because of the Russel paradox.

So the non-existence of the "set of all sets" is a consequence of the fact that presuming it's existence would lead to the contradiction described by Russel's paradox.

This is in fact the origin of Russel's paradox.

In his work "The Basic Laws of Arithmetic", Gottlob Frege had taken as a postulate the existence of this "set of all sets". In a letter to Frege, Bertrand Russell essentially blew away the basis of Frege's entire work by describing the paradox and proving that this postulate could not be a part of a consistent set theory.

This is how I had understood it before and I always found it to be the simplest answer, which is quite neat.
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Justin L.Jul 21 '10 at 5:35

3

If the "set of all sets" were to exist, because self-containment and non-self-containment are valid predicates, the set of all sets not containing themselves would have to exist as a set in order for our set theory to be consistent - Not true, as Quine's New Foundations, cited by François, shows. It has a universal set, but not the Russell set.
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Charles StewartJul 21 '10 at 11:05

1

I guess this depends on what one considers a "set theory". The usual image of the universe of sets, given by the cumulative hierarchy, doesn't allow for the set of all sets.
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Michael Greinecker♦Mar 8 '12 at 9:13

2

I don't see how you could not consider Quine's New Foundations a set theory; it has the usual universe of sets in it. It may in fact have a superset (superclass?) of ZF's sets, though it's possible stratified comprehension excludes some sets ZF includes. It is akin to NBG set theory; some definite weirdness if you're used to ZFC, but basically the same thing for practical purposes.
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prosfilaesOct 17 '12 at 6:48

NF doesn't necessarily have all of ZF's sets in it. While it's consistent that the von Neumann version of $\omega$ exists in NF, I believe it's not provable that it does.
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Malice VidrineJun 23 '14 at 10:12

It becomes paradoxical when you add the assumption that whenever $\varphi(x)$ is a formula, and $A$ is a preexisting set, then $\{x\in A\mid \varphi(x)\}$ is a set as well.

This is known as bounded comprehension, or separation. The full notion of comprehension was shown to be inconsistent by Russell's paradox. But this version is not so strikingly paradoxical. It is part of many of the modern axiomatizations of set theory, which have yet to be shown inconsistent.

We can show that assuming separation holds, the proof of the Russell paradox really translates to the following thing: If $A$ is a set, then there is a subset of $A$ which is not an element of $A$.

In the presence of a universal set this leads to an outright contradiction, because this subset should be an element of the set of all sets, but it cannot be.

But we may choose to restrict the formulas which can be used in this schema of axioms. Namely, we can say "not every formula should define a subset!", and that's okay. Quine defined a set theory called New Foundations, in which we limit these formulas in a way which allows a universal set to exist. Making it consistent to have the set of all sets, if we agree to restrict other parts of our set theory.

The problem is that the restrictions given by Quine are much harder to work with naively and intuitively. So we prefer to keep the full bounded comprehension schema, in which case the set of all set cannot exist for the reasons above.

While we are at it, perhaps it should be mentioned that the Cantor paradox, the fact that the power set of a universal set must be strictly larger, also fails in Quine's New Foundation for the same reasons. The proof of Cantor's theorem that the power set is strictly larger simply does not go through without using "forbidden" formulas in the process.

Not to mention that the Cantor paradox fails if we do not assume the power set axiom, namely it might be that not all sets have a power set. So if the universal set does not have a power set, there is no problem in terms of cardinality.

But again, we are taught from the start that these properties should hold for sets, and therefore they seem very natural to us. So the notion of a universal set is paradoxical for us, for that very reason. We are educated with a bias against universal sets. If you were taught that not all sets should have a power set, or that not all sub-collections of a set which are defined by a formula are sets themselves, then neither solution would be problematic. And maybe even you'd find it strange to think of a set theory without a universal set!

Can you expand a bit on "If A is a set, then there is a subset of A which is not an element of A." should it not be "If A is a set, then there is a set of subsets of A which is not an element of A."so suppose A = { 1,2,3,{1,2}} then i mean the set { {1,3} , {2,3} , {1, {1,2}} , {3,{1,2}} }
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WillemienJan 17 at 15:30

Yes, it could be also there is a set of subsets which is not an element of $A$. But there is also a subset of $A$ which is not an element of $A$, why is that a problem? There are also sets that has nothing to do with $A$ which are not elements of $A$.
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Asaf KaragilaJan 17 at 15:37

sorry I am just trying to understand it, could you give a me the subset you mean given the set A I gave?
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WillemienJan 17 at 15:45

Well, $A$ itself is that subset in the case of $A=\{1,2,3,\{1,2\}\}$. It is true, $\varnothing$ could also work, and so could any singleton, or so on. But given any set, $A$ the set $\{x\in A\mid x\notin x\}$ is either not a set, or not an element of $A$. In presence of comprehension axioms which do not restrict the formulas, it is always a set, and therefore not an element of $A$. In Quine's New Foundations, and other more "type theoretic" set theories, we restrict the formulas that can define a set, which mean it might not be a set at all.
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Asaf KaragilaJan 17 at 15:54

Russel's paradox arises if you consider the set $U=\left\{x:x\not\in x\right\}$. Ask yourself if $U\in U$. If you suppose so, then by the definition of unrestricted set comprehension $U\not\in U$. You have a contradiction, so it must be the opposite of what you supposed, that is, $U\not\in U$. But this is the same as saying $U$ belongs to the complement of itself, that is, $U\in U$. You now have another contradiction, but this is far worse, since you have no hypotheses. The whole theory is logically inconsistent.

In set theory there are two ways for getting rid of the Russel's paradox: either you disallow the set of all sets and other similar sets (see for example the Zermelo-Fraenkel set theory), or you allow them, but you also restrict the way they are used (see for example the Morse-Kelley set theory).

In the first case, set comprehension says if you have a set $A$ you can have $\left\{x\in A:\phi\left(x\right)\right\}$ (notice: writing $\left\{x:\phi\left(x\right)\right\}$ is just wrong in this case, because you have to have an initial set). If you now define $U=\left\{x\in A:x\not\in x\right\}$ and you repeat the same passages as before, it only follows that $U\not\in A$. There's no contradiction and the theory is consistent.

In the second case, you consider classes, not just sets. Sets are classes that belong to some other class, while proper classes are classes that belong to no class. Set comprehension, in this case, says you can have $\left\{x:\phi\left(x\right)\right\}$, but all its members are sets by definition. If try to reproduce Russel's paradox, you get that $U\not\in U$. If you then suppose that $U$ is a set, then you have a contradiction, so $U$ must be a proper class. This is all you get. No contradictions. The theory is consistent.

There are other ways to avoid Russell's paradox, too. For example, the axiom schema of comprehension in NF doesn't allow you to construct $\{x: x \notin x\}$ (or, AIUI, even $\{x \in A: x \notin x\}$ in general) because $x \notin x$ is not a stratified formula. It does let you construct the universal set $\{x: x=x\}$, though.
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Ilmari KaronenMar 8 '12 at 8:57

An informal explanation is Russel's Paradox. The wiki page is informative, here's the relevant quote:

Let us call a set "abnormal" if it is
a member of itself, and "normal"
otherwise. For example, take the set
of all squares. That set is not itself
a square, and therefore is not a
member of the set of all squares. So
it is "normal". On the other hand, if
we take the complementary set that
contains all non-squares, that set is
itself not a square and so should be
one of its own members. It is
"abnormal".

Now we consider the set of
all normal sets, R. Attempting to
determine whether R is normal or
abnormal is impossible: If R were a
normal set, it would be contained in
the set of normal sets (itself), and
therefore be abnormal; and if it were
abnormal, it would not be contained in
the set of normal sets (itself), and
therefore be normal. This leads to the
conclusion that R is both normal and
abnormal: Russell's paradox.

I mentioned this paradox in my question; it however is not the same one that I am asking about.
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Justin L.Jul 20 '10 at 23:17

2

You're right, sorry. Probably you can say that, since "the set of all sets" is not so well-defined, it must necessarily contain the set from Russel's paradox. Since the set from Russel's Paradox can't exist, neither can any set that contains it. But I feel this is cheating somehow.
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Edan MaorJul 20 '10 at 23:37

That's not cheating, that's exactly correct. I always use Russel's Paradox for the "set of all sets" question. The main point in both cases anyway is remembering that Cantor's definition of a set is a little to vague if you dig deep enough.
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balpha♦Jul 21 '10 at 5:10

You can still make the cardinality argument that the set cannot contain all its subsets.
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Michael Greinecker♦Mar 8 '12 at 9:11

Insightful! @MichaelGreinecker. In type theory, the set of all the subsets resides on upstairs.
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George ChenAug 22 '14 at 23:50

A set cannot have any of its subset as members because a subset of a set involves the totality also, and therefore must not be a member of the larger set; otherwise there will be a vicious circle.
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George ChenAug 23 '14 at 0:42

1

@GeorgeChen There is something in set theory called a transitive set, and for such a set, every element is a subset. Here is an example: $\{\emptyset\}$.
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Michael Greinecker♦Aug 23 '14 at 7:25

This is wild. In type theory, the subclass stays on the same floor; the member goes to downstairs. They are totally different things.
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George ChenAug 23 '14 at 9:02

When set theory was invented ir was assumed that given any predicate, there was a set containing all the things that satisfied the predicate. This assumption is called naive comprehension. Unfortunately this allowed paradoxes like the set of all sets not containing themselves/

So people invented restricted axioms of comprehension. These are rules that say that only certain predicates give rise to sets. There are different kinds of set theory with different comprehension restrictions.

One idea for limiting comprehension is to say that if the class of things satisfies the predicate is too big, the class is not a set. All the commonly used set theories use this idea. The set of all sets is very large indeed, so it is not a set in these theories.

There are other ways of restricting comprehension that get rid of the paradoxical sets but do allow a universal set. One such set theory was invented by Quine and called New Foundations. There is a book by Holmes, Set Theory with a Universal Set.

"Why is "the set of all sets" a
paradox? It seems like it would be
fine, to me."

The lesson of this and other set theory paradoxes is that when you define a set in a proof, you are obliged to demonstrate that the set exists and is well-defined. To me, a paradox of Russell's Paradox is that he fell into this trap since he pretty much wore himself out trying to generate a rigorous foundation for Mathematics.

The set of all sets is well-defined in the set theory Cantor used: to any logical predicate there is a set of all objects satisfying it. The problem is that Cantor's set theory is inconsistent.
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HurkylMar 8 '12 at 9:36

I think this paradoxon is not so complicated and can be explained simply.

If you put all existing sets S(1) to S(k) together in a set S(all), you create a new set S(k+1)={S(1),...,S(k)} and this set has to be in the set of all sets as well (because it should contain all sets). Thus, you put the new set S(k+1) in the set of all sets S(all). But this in turn creates a new set S(k+2)={S(1),...,S(k+1)} wich has to be inserted in the set of all sets as well. Putting S(k+2) in S(all), however, creates a new set and so on and so on.

In conclusion, everytime you put the new set into the set of all sets you create a new set. As a consequence, you can never build a set S(all) that contains all sets without creating a set which is not contained in S(all).

Building a set inductively like this is not paradoxical. The natural numbers are built in exactly this way starting with the empty set. You can't talk about 'putting a set in' to the set of all sets.
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Daniel RustMay 3 at 22:56

The building process is not the paradoxon, but it shows that a set cannot contains all sets without producing a new set that is not contained in this set and hence a set that is assumed to contain all sets actually does not contain all sets. The latter is the paradoxon.
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user237596May 4 at 1:05

But ok, I am not a mathematician and this solution was the first that comes to my mind. Maybe, it is completely bullshit ;-)
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user237596May 4 at 1:11

I am just curious. You mentioned the natural numbers. In my opinion, you have a similar case if you say there is a natural number that is maximal. You can simply show that this number cannot be maximal because you can add +1 to this number. Isn't this case similar?
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user237596May 4 at 1:16

with "there is a natural number that is maximal" I meant the assumption that there is a natural number that is greater than all other natural numbers. 1) set that contains all set => this set is larger than all other sets 2) maximal number => this number is greater than all other natural numbers. I see there some similarities.
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user237596May 4 at 2:23

Russell's paradox, statement in set theory, devised by the English mathematician-philosopher Bertrand Russell , that demonstrated a flaw in earlier efforts to axiomatize the subject.

Russell found the paradox in 1901 and communicated it in a letter to the German mathematician-logician Gottlob Frege in 1902. Russell's letter demonstrated an inconsistency in Frege's axiomatic system of set theory by deriving a paradox within it. (The German mathematician Ernest Zermelo had found the same paradox independently; since it could not be produced in his own axiomatic system of set theory, he did not publish the paradox.)

Frege had constructed a logical system employing an unrestricted comprehension principle. The comprehension principle is the statement that, given any condition expressible by a formula ϕ(x), it is possible to form the set of all sets x meeting that condition, denoted {x | ϕ(x)}. For example, the set of all sets—the universal set—would be {x | x = x}.

It was noticed in the early days of set theory, however, that a completely unrestricted comprehension principle led to serious difficulties. In particular, Russell observed that it allowed the formation of {x | x ∉ x}, the set of all non-self-membered sets, by taking ϕ(x) to be the formula x ∉ x. Is this set—call it R—a member of itself? If it is a member of itself, then it must meet the condition of its not being a member of itself. But if it is not a member of itself, then it precisely meets the condition of being a member of itself. This impossible situation is called Russell's paradox.

The significance of Russell's paradox is that it demonstrates in a simple and convincing way that one cannot both hold that there is meaningful totality of all sets and also allow an unfettered comprehension principle to construct sets that must then belong to that totality. (Russell spoke of this situation as a “vicious circle.”)

Set theory avoids this paradox by imposing restrictions on the comprehension principle. The standard Zermelo-Fraenkel axiomatization does not allow comprehension to form a set larger than previously constructed sets. (The role of constructing larger sets is given to the power-set operation.) This leads to a situation where there is no universal set—an acceptable set must not be as large as the universe of all sets.

A very different way of avoiding Russell's paradox was proposed in 1937 by the American logician Willard Van Orman Quine. In his paper “New Foundations for Mathematical Logic,” the comprehension principle allows formation of {x | ϕ(x)} only for formulas ϕ(x) that can be written in a certain form that excludes the “vicious circle” leading to the paradox. In this approach, there is a universal set.

Your answer as originally written was blatant plagiarism. I have improved it so that now it simply does not answer the question and is a bit condescending. You should strive for higher goals.
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Eric StuckyJun 23 '14 at 10:04